MEN OF MATHEMATICS tion when he mistakenly imagined he had succeeded. Through Holmboe the supposed solution was sent to the most learned mathematical scholar of the time in Denmark who, fortunately for Abel, asked for further particulars without committing him- self to an opinion on the correctness of the solution. Abel in the meantime had found the flaw in his reasoning. The supposed solution was of course no solution at all This failure gave him a most salutary jolt; it jarred him on to the right track and caused him to doubt whether an algebraic solution was possible. He proved the impossibility. At the time he was about nineteen. But he had been anticipated, at least in part, in the whole project. As this question of the general quintic played a role hi algebra sirnifar to that of a crucial experiment to decide the fate of an entire scientific theory, it is worth a moment's attention. We shall quote presently a few things Abel Mm self says. The nature of the problem is easily described. In early school algebra we learn to solve the general equations of the first and second degrees in the unknown #, say ax 4- b = 0, ox- -r bx -f- c = 0, and a little later those of the third and fourth degrees, say That is, we produce finite (closed) formulae for each of these general equations of the first four degrees, expressing the unknown x in terms of the given coefficients a9b9c,d,e. A solution such as any one of these four which can be obtained by only a finite number ofaddit ions, multiplications, subtractions, divisions, and extractions of roots, aH these operations being performed on the given coefficients, is called algebraic. The important qualifi- cation in this definition of an algebraic solution is 'finite' ; there is no difficulty in describing solutions for any algebraic equation which contain no extractions of roots at all, but which do imply an infinity of the other operations named. After this success with algebraic equations of the first four 840